Entanglement patterns in mutually unbiased basis sets

Abstract

A few simply stated rules govern the entanglement patterns that can occur in mutually unbiased basis sets (MUBs) and constrain the combinations of such patterns that can coexist in full complements of MUBs. We consider Hilbert spaces of prime power dimensions ($D={p}^{N}$), as realized by systems of $N$ prime-state particles, where full complements of $D+1$ MUBs are known to exist, and we assume only that MUBs are eigenbases of generalized Pauli operators, without using any particular construction. The general rules include the following: (1) In any MUB, a given particle appears either in a pure state or totally entangled and (2) in any full MUB complement, each particle is pure in ($p+1$) bases (not necessarily the same ones) and totally entangled in the remaining (${p}^{N}\ensuremath{-}p$). It follows that the maximum number of product bases is $p+1$ and, when this number is realized, all remaining (${p}^{N}\ensuremath{-}p$) bases in the complement are characterized by the total entanglement of every particle. This ``standard distribution'' is inescapable for two-particle systems (of any $p$), where only product and generalized Bell bases are admissible MUB types. This and the following results generalize previous results for qubits [Phys. Rev. A 65. 032320 (2002); Phys. Rev. A 72, 062310 (2005)] and qutrits [Phys. Rev. A 70, 012302 (2004)], drawing particularly upon [Phys. Rev. A 72, 062310 (2005)]. With three particles there are three MUB types, and these may be combined in ($p+2$) different ways to form full complements. With $N=4$, there are 6 MUB types for $p=2$, but new MUB types become possible with larger $p$, and these are essential to realizing full complements. With this example, we argue that new MUB types that show new entanglement patterns should enter with every step in $N$ and, also, when $N$ is a prime plus 1, at a critical $p$ value, $p=N\ensuremath{-}1$. Such MUBs should play critical roles in filling complements.